FOM: Goading Harvey

[Joseph Shoenfield]

     Steve has completely misinterpreted my motives in my discussion of 
the possibility of Harvey's work leading to a theorem somewhat resembling 
the Martin-Steel Theorem.   I did want to goad Harvey into working on 
such a result.   This is because Harvey is the only person (in my opinion) 
with both the knowledge and the mathematical ability needed to prove such 
a result, and because I think such a result would fit in very well with 
his program.   I had no intention of comparing him unfavorably with Martin 
or Steel.   They are certainly three of the outstanding logicians of today.
I see no reason for me to try to compare them; if I did, I would not disguise
this comparison in the way that Steve suggests I have.   As to the Karp prize,
I mentioned it only to illustrate the large splash that the Martin-Steel
theorem has made in logic.   I certainly did not think Harvey would be 
tempted by the meager financial rewards.   Harvey has had many successes
in logic; but I think it is clear that the particular result which has 
inspired this debate has not and will not create a similar splash.
    I haven't thought nearly enough about the matter to make a conjecture;
but I will state a possible theorem which I hope will clarify what I have 
in mind: There is a sequence {An} of sentences such that An is strictly 
pi-1-n+1, and An can be proved from ZFC and the existence of n large 
cardinals of a certain sort but not (under obvious consistency hypotheses) 
from n-1 such cardinals.   The kind of large cardinal remains to be 
determined.   (Recall that the definition of a Woodin cardinal was an 
essential preliminary to the Martin-Steel theorem; this was part of the 
reason that Woodin shared in the prize.)   One would of course hope to 
show that one could not replace An by a pi-1-n sentence.   It would be 
nice if An was simple in some of the ways that undecidable sentences 
considered by Harvey and others are simple; but I do not think this 
essential for the significance of such a result.